1. Spring Forces and Simple Harmonic Motion
Chapter 10

2. Expectations At the end of this chapter, students will be able to:
Apply Hooke’s Law to the calculation of spring forces.
Conceptually understand how simple harmonic motion is caused by forces and torques obeying Hooke’s Law.
Calculate displacements, velocities, accelerations, and frequencies for objects undergoing simple harmonic motion.

3. Expectations At the end of this chapter, students will be able to:
Calculate the elastic potential energy resulting from work done by spring forces.
Understand how pendulums approximate simple harmonic motion.
Calculate the natural frequencies of both simple and physical pendulums.
Conceptually understand the ideas of damped, driven, and resonant simple harmonic motion.

4. Expectations At the end of this chapter, students will be able to:
Analyze the elastic deformation of objects in terms of stress, strain, and the elastic moduli of materials.
Express Hooke’s Law in terms of stress and strain.

5. Spring Forces
A spring resists being stretched or compressed.

6. Spring Forces
The force with which the spring resists is proportional to the distance through which it is compressed or stretched:
“spring constant”
SI units: N/m

7. Spring Forces – Hooke’s Law
This relationship is called Hooke’s Law.

8. Hooke’s Law Robert Hooke 1635 – 1703
English mathematician and natural philosopher; contemporary of Isaac Newton
Early builder of microscopes and telescopes

9. Hooke’s Law
The direction of the spring force is always opposite the direction of the stretching or compression of the spring – hence, the minus sign.

10. A Consequence of Hooke’s Law
Consider Newton’s second law and Hooke’s Law simultaneously:
Now, a small and sneaky bit of calculus:
differential equation
its solution

11. Simple Harmonic Motion
Motion described by this equation:
is called simple harmonic motion (SHM). It is:
periodic (repeats itself in time)
oscillatory (takes place over a limited spatial range)
angular frequency (rad/s)
displacement (m)
time (s)
amplitude (m)

12. Simple Harmonic Motion
A = 1.00 m

13. SHM: Reference Circle Representation
A vector of magnitude
A rotates about the
origin with an angular
velocity w.
The x component of
the vector represents
the displacement.

14. SHM: Frequency
Since there are 2p radians in each trip (“cycle”) around the reference circle, the “cycle” frequency is related to the angular frequency by
SI units of “cycle” frequency, f:
cycles / s = Hertz (Hz)

15. SHM: Velocity We can calculate the velocity from the reference circle
representation:

16. SHM: Acceleration

17. SHM: Velocity and Acceleration
w must be expressed in rad/s.
Like displacement, velocity and acceleration are periodic in time.
Maximum velocity:
Maximum acceleration:
Acceleration has maximum magnitude at extremes of displacement.
Velocity has maximum magnitude when displacement is zero.

18. Mass on a Spring System
Natural frequency for a mass m on a spring with spring constant k:

19. Work Done in Straining a Spring
Stretch or compress a spring by a displacement x from its unstrained length.
Initial force: F0 = Final force: Fmax = kx
Average force:
Work over a displacement x:

20. Elastic Potential Energy
The spring force is a conservative force:
Like all conservative forces, its work is path-independent.
Like all conservative forces, it is associated with a form of stored or potential energy.
Elastic potential energy:

21. Total Mechanical Energy
We add another term:

22. The Simple Pendulum
A simple pendulum is a particle attached to one end of a massless cord of length L. It is able to swing freely and without friction from the other end of the cord.
Its frequency:

23. The Physical Pendulum
A physical pendulum is any real object (mass m) suspended a distance L from its center of gravity, able to swing freely and without friction from the suspension point.
Its frequency:

24. Physical-Simple Correspondence
Notice that a simple pendulum would have a moment of inertia:
Substitute:
As a physical pendulum becomes a simple one, its frequency “collapses” to that of a simple pendulum.

25. Small-Angle Approximation
The restoring torque on a pendulum does not actually have the Hooke’s Law form:
Result: the restoring torque increases with angle, but at less than a linear rate.

26. Small-Angle Approximation
How much less?
q, °
% under linear
0.10
0.0002%
1.0
0.02%
2.0
0.08%
5.0
0.51% 10
2.0%

27. Damped Oscillations
If the only force doing work on an object is the spring force (conservative), its mechanical energy is conserved. If frictional forces also do work, the object’s mechanical energy decreases, and the SHM is called damped.
If the frictional force is just large enough to prevent oscillation as the object reaches its equilibrium position, it is called critically damped.

28. Driven Oscillations
If a driving force acts on an object in addition to a Hooke’s Law restoring force, the harmonic motion of the object is called driven.
Example: a tree in a gusty wind.

29. Driven Oscillations: Resonance
If the driving force is periodic, and is applied at the natural frequency of the oscillating object, the work done on the object adds up over multiple cycles of motion, and large-amplitude motion results. This is called resonance. The natural frequency is sometimes called the resonant frequency.
Example: a person on a swing, being pushed by another person.

30. Material Deformation: Everything is a Spring
Solid materials are interconnected, microscopically, by powerful intermolecular bonding forces.
These forces behave like springs … with really large spring constants.
Because of them, material objects resist deformations, such as compression, elongation, or shearing.

31. Tension and Compression
A force acts to increase the length of an object:
fractional change in length
applied force
cross-
sectional
area
Young’s modulus
SI units = N/m2

32. Thomas Young 1773 - 1829 English physicist, physician, and
Egyptologist
Famous mostly for his
work in optics

33. Shear A pair of forces act to shear an object (deform it slantwise):
applied force
cross-sectional
area
Shear modulus
SI units = N/m2

34. Shear 1945 - ? Has only one name English physicist and pop musician
Inventor of the shear
modulus
Rumored to have appeared in the 1983 version of Dune

35. Volume Deformation
In order to discuss volume deformation, it is necessary to define a new force-related quantity: pressure.
Pressure is the ratio of the magnitude of a force applied perpendicular to a surface to the area of that surface:
SI units: N/m2 = Pascals (Pa)

36. Blaise Pascal 1623 – 1662 French mathematician
Invented the first digital
calculator (the “Pascaline”)

37. Volume Deformation
A change in pressure changes the volume of an object:
pressure change
fractional
change in
volume
bulk modulus
SI units: N/m2

38. Stress and Strain
Stress is the deforming force applied to an object, divided by its cross-sectional area:
Stress has SI units of N/m2 (just as pressure and the elastic moduli have).

39. Stress and Strain
Strain is the change in a dimensional quantity expressed as a fraction of its un-deformed value:
Strain is a dimensionless, unitless ratio.
Strain is the result of stress on a material object.

40. Stress and Strain Consider the defining equation for Young’s modulus:
Rearrange:
To restate:
This is the stress-strain formulation of Hooke’s Law.

41. Summary: Elastic Moduli
deformation elastic modulus equation
length Young’s modulus (Y)
shear shear modulus (S)
volume bulk modulus (B)
all moduli have the same SI units: N/m2 = Pa